A mathematical explanation of a 3-legged stool’s complete inability to wobble

By Abraham Piper

I imagine the title of this post has already driven away the folks who don’t have time to be curious about the chair they’re sitting in right now. But for the rest of us…

In a 2001 conversation on The Math Forum, a student, Teri, asked Dr. Ian why a 3-legged stool is always steady, but a 4-legged stool can be wobbly.

Thankfully for all of us eagerly reading 10 years later, Dr. Ian was more than happy to break it down for her…

Think of it this way:

(1) If you hold a cane in the air, you can move it in any direction,  twirl it, and so on. Its motion isn’t constrained at all. That is, the top of the cane can move freely in three dimensions.

(2) If you put (and keep) one end on the ground, now its motion is constrained: you can’t lift it, or rotate it… although you can swing the top around in a variety of different arcs. That is, the top of the cane can move freely in two dimensions.

(3) If you connect the tops of two canes together and place the other ends on the ground, you can still move the tops, but only along a single (straight) arc, back and forth. That is, the tops of the canes can move freely in one dimension.

(4) If you try the same trick with three canes, now you can’t move the tops at all. This is basically what’s happening with a three-legged stool. The tops of the cans can move in zero dimensions…which is to say, they can’t.

Each time you add a cane, you remove one dimension in which the top can move freely – that is, each new cane removes one ‘degree of freedom’.

Now, what happens when you add a fourth cane? Well, now you have too many constraints. This means that there are multiple ways that the stool can ‘solve’ the problem of which legs to use for support.…

In effect, during the time that the stool is actually wobbling, it’s really a two-legged stool, with one degree of freedom – which is the direction of the wobble.

He then goes on and gets more mathy than I can follow. But I love how he closes his response…

Write back if you’d like to talk about this some more…

As a matter of fact Teri did want to talk more about this most riveting of topics.

She followed up with…

[T]he consensus [among furniture makers] seems to be the more legs, the more stability. This concept seems at odds with your analysis. Can you explain this for me?

And the answer is Yes. Yes, Dr. Ian can explain that for you, Teri…

There are different kinds of stability! A three-legged stool is guaranteed not to wobble, because the ends of its legs always form a plane.…

More important for practical purposes, such a stool is LESS stable than one with more legs in the sense that its center of gravity is further inside its base: the more sides a regular polygon has, the greater its apothem (the distance from the center to the middle of an edge).

That greater distance means that the sitter can lean farther out in any direction without tipping over. So if you don’t mind a little tipping but don’t want to fall on your face, or if you have a reasonably even floor, more legs are better. In a barn, I’ll take the three-legged stool over the upholstered desk chair any time!

There you have it. Now go enjoy your day safely entrenched in the ample apothem of your wobbly desk chair.

(via Book of Joe)

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